Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity

TL;DR

This paper introduces high-order finite volume schemes that exactly preserve known equilibrium solutions of the Euler equations with gravity, improving accuracy in simulations involving both steady and dynamic states.

Contribution

It develops a novel high-order well-balanced finite volume scheme for Euler equations with gravity, capable of exactly maintaining prescribed equilibrium solutions.

Findings

Scheme achieves up to fifth order accuracy in 1D.

Method effectively preserves equilibrium states in numerical tests.

Applicable to non-steady problems with shocks.

Abstract

This work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is known, and we construct a scheme which is exactly well balanced for that particular equilibrium. The scheme is based on high order reconstructions of the fluctuations from equilibrium of density, momentum and pressure, and on a well balanced integration of the source terms, while no assumptions are needed on the numerical flux, beside consistency. This technique allows to construct well balanced methods also for a class of moving equilibria. Several numerical tests demonstrate the performance of the scheme on different scenarios, from equilibrium solutions to non steady problems involving shocks. The numerical tests are carried out with methods up to fifth order in one dimension, and third order accuracy in 2D.